We study the related problems of denoising images corrupted by impulsive noise and blind inpainting (i.e., inpainting when the deteriorated region is unknown). Our basic approach is to model the set of patches of pixels in an image as a union of low dimensional subspaces, corrupted by sparse but perhaps large magnitude noise. For this purpose, we develop a robust and iterative RANSAC like method for single subspace modeling and extend it to an iterative algorithm for modeling multiple subspaces. We prove convergence for both algorithms and carefully compare our methods with other recent ideas for such robust modeling. We demonstrate state of the art performance of our method for both imaging problems.

Abstract. We describe a method for fast approximation of sparse coding. A given input vector is passed through a binary tree. Each leaf of the tree contains a subset of dictionary elements. The coefﬁcients corresponding to these dictionary elements are allowed to be nonzero and their values are calculated quickly by multiplication with a precomputed pseudoinverse. The tree parameters, the dictionary, and the subsets of the dictionary corresponding to each leaf are learned. In the process of describing this algorithm, we discuss the more general problem of learning the groups in group structured sparse modeling. We show that our method creates good sparse representations by using it in the object recognition framework of [1,2]. Implementing our own fast version of the SIFT descriptor the whole system runs at 20 frames per second on 321 × 481 sized images on a laptop with a quad-core cpu, while sacriﬁcing very little accuracy on the Caltech 101, Caltech 256, and 15 scenes benchmarks.